LMFDB - Embedded newform 7800.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$62.2833135766$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.37228 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.37228 q^{7} +1.00000 q^{9} -3.37228 q^{11} +1.00000 q^{13} +1.37228 q^{17} -6.74456 q^{19} -3.37228 q^{21} +0.627719 q^{23} +1.00000 q^{27} -2.00000 q^{29} +6.74456 q^{31} -3.37228 q^{33} -5.37228 q^{37} +1.00000 q^{39} -1.37228 q^{41} +4.00000 q^{43} +1.25544 q^{47} +4.37228 q^{49} +1.37228 q^{51} +9.37228 q^{53} -6.74456 q^{57} -8.00000 q^{59} +8.11684 q^{61} -3.37228 q^{63} +4.00000 q^{67} +0.627719 q^{69} -11.3723 q^{71} +15.4891 q^{73} +11.3723 q^{77} -16.8614 q^{79} +1.00000 q^{81} +12.0000 q^{83} -2.00000 q^{87} +13.3723 q^{89} -3.37228 q^{91} +6.74456 q^{93} -2.62772 q^{97} -3.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - q^7 + 2 * q^9	$ 2 q + 2 q^{3} - q^{7} + 2 q^{9} - q^{11} + 2 q^{13} - 3 q^{17} - 2 q^{19} - q^{21} + 7 q^{23} + 2 q^{27} - 4 q^{29} + 2 q^{31} - q^{33} - 5 q^{37} + 2 q^{39} + 3 q^{41} + 8 q^{43} + 14 q^{47} + 3 q^{49} - 3 q^{51} + 13 q^{53} - 2 q^{57} - 16 q^{59} - q^{61} - q^{63} + 8 q^{67} + 7 q^{69} - 17 q^{71} + 8 q^{73} + 17 q^{77} - 5 q^{79} + 2 q^{81} + 24 q^{83} - 4 q^{87} + 21 q^{89} - q^{91} + 2 q^{93} - 11 q^{97} - q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - q^7 + 2 * q^9 - q^11 + 2 * q^13 - 3 * q^17 - 2 * q^19 - q^21 + 7 * q^23 + 2 * q^27 - 4 * q^29 + 2 * q^31 - q^33 - 5 * q^37 + 2 * q^39 + 3 * q^41 + 8 * q^43 + 14 * q^47 + 3 * q^49 - 3 * q^51 + 13 * q^53 - 2 * q^57 - 16 * q^59 - q^61 - q^63 + 8 * q^67 + 7 * q^69 - 17 * q^71 + 8 * q^73 + 17 * q^77 - 5 * q^79 + 2 * q^81 + 24 * q^83 - 4 * q^87 + 21 * q^89 - q^91 + 2 * q^93 - 11 * q^97 - q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.37228	−1.27460	−0.637301	−	0.770615i	$-0.719949\pi$
$7$	−3.37228	−1.27460	−0.637301	+	0.770615i	$0.719949\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.37228	−1.01678	−0.508391	−	0.861127i	$-0.669759\pi$
$11$	−3.37228	−1.01678	−0.508391	+	0.861127i	$0.669759\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.37228	0.332827	0.166414	−	0.986056i	$-0.446781\pi$
$17$	1.37228	0.332827	0.166414	+	0.986056i	$0.446781\pi$
$18$	0	0
$19$	−6.74456	−1.54731	−0.773654	−	0.633608i	$-0.781573\pi$
$19$	−6.74456	−1.54731	−0.773654	+	0.633608i	$0.781573\pi$
$20$	0	0
$21$	−3.37228	−0.735892
$22$	0	0
$23$	0.627719	0.130888	0.0654442	−	0.997856i	$-0.479154\pi$
$23$	0.627719	0.130888	0.0654442	+	0.997856i	$0.479154\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	6.74456	1.21136	0.605680	−	0.795709i	$-0.292901\pi$
$31$	6.74456	1.21136	0.605680	+	0.795709i	$0.292901\pi$
$32$	0	0
$33$	−3.37228	−0.587039
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.37228	−0.883198	−0.441599	−	0.897213i	$-0.645589\pi$
$37$	−5.37228	−0.883198	−0.441599	+	0.897213i	$0.645589\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−1.37228	−0.214314	−0.107157	−	0.994242i	$-0.534175\pi$
$41$	−1.37228	−0.214314	−0.107157	+	0.994242i	$0.534175\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.25544	0.183124	0.0915622	−	0.995799i	$-0.470814\pi$
$47$	1.25544	0.183124	0.0915622	+	0.995799i	$0.470814\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	1.37228	0.192158
$52$	0	0
$53$	9.37228	1.28738	0.643691	−	0.765286i	$-0.277402\pi$
$53$	9.37228	1.28738	0.643691	+	0.765286i	$0.277402\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.74456	−0.893339
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	8.11684	1.03926	0.519628	−	0.854393i	$-0.326071\pi$
$61$	8.11684	1.03926	0.519628	+	0.854393i	$0.326071\pi$
$62$	0	0
$63$	−3.37228	−0.424868
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0.627719	0.0755684
$70$	0	0
$71$	−11.3723	−1.34964	−0.674821	−	0.737982i	$-0.735779\pi$
$71$	−11.3723	−1.34964	−0.674821	+	0.737982i	$0.735779\pi$
$72$	0	0
$73$	15.4891	1.81286	0.906432	−	0.422351i	$-0.138795\pi$
$73$	15.4891	1.81286	0.906432	+	0.422351i	$0.138795\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.3723	1.29599
$78$	0	0
$79$	−16.8614	−1.89706	−0.948528	−	0.316693i	$-0.897428\pi$
$79$	−16.8614	−1.89706	−0.948528	+	0.316693i	$0.897428\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	13.3723	1.41746	0.708729	−	0.705480i	$-0.249269\pi$
$89$	13.3723	1.41746	0.708729	+	0.705480i	$0.249269\pi$
$90$	0	0
$91$	−3.37228	−0.353511
$92$	0	0
$93$	6.74456	0.699379
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.62772	−0.266804	−0.133402	−	0.991062i	$-0.542590\pi$
$97$	−2.62772	−0.266804	−0.133402	+	0.991062i	$0.542590\pi$
$98$	0	0
$99$	−3.37228	−0.338927
$100$	0	0
$101$	4.74456	0.472102	0.236051	−	0.971741i	$-0.424147\pi$
$101$	4.74456	0.472102	0.236051	+	0.971741i	$0.424147\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.62772	0.834073	0.417037	−	0.908890i	$-0.363069\pi$
$107$	8.62772	0.834073	0.417037	+	0.908890i	$0.363069\pi$
$108$	0	0
$109$	8.74456	0.837577	0.418789	−	0.908084i	$-0.362455\pi$
$109$	8.74456	0.837577	0.418789	+	0.908084i	$0.362455\pi$
$110$	0	0
$111$	−5.37228	−0.509914
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−4.62772	−0.424222
$120$	0	0
$121$	0.372281	0.0338438
$122$	0	0
$123$	−1.37228	−0.123734
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.2337	1.44051	0.720253	−	0.693711i	$-0.244026\pi$
$127$	16.2337	1.44051	0.720253	+	0.693711i	$0.244026\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	−18.7446	−1.63772	−0.818860	−	0.573993i	$-0.805394\pi$
$131$	−18.7446	−1.63772	−0.818860	+	0.573993i	$0.805394\pi$
$132$	0	0
$133$	22.7446	1.97220
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.4891	−0.981582	−0.490791	−	0.871277i	$-0.663292\pi$
$137$	−11.4891	−0.981582	−0.490791	+	0.871277i	$0.663292\pi$
$138$	0	0
$139$	7.37228	0.625309	0.312654	−	0.949867i	$-0.398782\pi$
$139$	7.37228	0.625309	0.312654	+	0.949867i	$0.398782\pi$
$140$	0	0
$141$	1.25544	0.105727
$142$	0	0
$143$	−3.37228	−0.282004
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.37228	0.360620
$148$	0	0
$149$	13.3723	1.09550	0.547750	−	0.836642i	$-0.315485\pi$
$149$	13.3723	1.09550	0.547750	+	0.836642i	$0.315485\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	1.37228	0.110942
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	9.37228	0.743270
$160$	0	0
$161$	−2.11684	−0.166831
$162$	0	0
$163$	0.627719	0.0491667	0.0245834	−	0.999698i	$-0.492174\pi$
$163$	0.627719	0.0491667	0.0245834	+	0.999698i	$0.492174\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.74456	−0.515770
$172$	0	0
$173$	−7.48913	−0.569388	−0.284694	−	0.958618i	$-0.591892\pi$
$173$	−7.48913	−0.569388	−0.284694	+	0.958618i	$0.591892\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	24.2337	1.81131	0.905655	−	0.424014i	$-0.139379\pi$
$179$	24.2337	1.81131	0.905655	+	0.424014i	$0.139379\pi$
$180$	0	0
$181$	−5.37228	−0.399319	−0.199659	−	0.979865i	$-0.563984\pi$
$181$	−5.37228	−0.399319	−0.199659	+	0.979865i	$0.563984\pi$
$182$	0	0
$183$	8.11684	0.600014
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.62772	−0.338412
$188$	0	0
$189$	−3.37228	−0.245297
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	−9.37228	−0.674632	−0.337316	−	0.941392i	$-0.609519\pi$
$193$	−9.37228	−0.674632	−0.337316	+	0.941392i	$0.609519\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.2337	0.729120	0.364560	−	0.931180i	$-0.381219\pi$
$197$	10.2337	0.729120	0.364560	+	0.931180i	$0.381219\pi$
$198$	0	0
$199$	21.4891	1.52332	0.761662	−	0.647975i	$-0.224384\pi$
$199$	21.4891	1.52332	0.761662	+	0.647975i	$0.224384\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	6.74456	0.473375
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.627719	0.0436295
$208$	0	0
$209$	22.7446	1.57327
$210$	0	0
$211$	22.9783	1.58189	0.790944	−	0.611889i	$-0.209590\pi$
$211$	22.9783	1.58189	0.790944	+	0.611889i	$0.209590\pi$
$212$	0	0
$213$	−11.3723	−0.779216
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−22.7446	−1.54400
$218$	0	0
$219$	15.4891	1.04666
$220$	0	0
$221$	1.37228	0.0923096
$222$	0	0
$223$	−18.9783	−1.27088	−0.635439	−	0.772151i	$-0.719181\pi$
$223$	−18.9783	−1.27088	−0.635439	+	0.772151i	$0.719181\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.7446	1.24412	0.622060	−	0.782969i	$-0.286296\pi$
$227$	18.7446	1.24412	0.622060	+	0.782969i	$0.286296\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	11.3723	0.748241
$232$	0	0
$233$	18.6277	1.22034	0.610171	−	0.792270i	$-0.291101\pi$
$233$	18.6277	1.22034	0.610171	+	0.792270i	$0.291101\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−16.8614	−1.09527
$238$	0	0
$239$	−22.3505	−1.44574	−0.722868	−	0.690986i	$-0.757176\pi$
$239$	−22.3505	−1.44574	−0.722868	+	0.690986i	$0.757176\pi$
$240$	0	0
$241$	−24.9783	−1.60899	−0.804495	−	0.593959i	$-0.797564\pi$
$241$	−24.9783	−1.60899	−0.804495	+	0.593959i	$0.797564\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.74456	−0.429146
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	17.4891	1.10390	0.551952	−	0.833876i	$-0.313883\pi$
$251$	17.4891	1.10390	0.551952	+	0.833876i	$0.313883\pi$
$252$	0	0
$253$	−2.11684	−0.133085
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.0000	−0.623783	−0.311891	−	0.950118i	$-0.600963\pi$
$257$	−10.0000	−0.623783	−0.311891	+	0.950118i	$0.600963\pi$
$258$	0	0
$259$	18.1168	1.12573
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	17.4891	1.07843	0.539213	−	0.842170i	$-0.318722\pi$
$263$	17.4891	1.07843	0.539213	+	0.842170i	$0.318722\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	13.3723	0.818370
$268$	0	0
$269$	12.7446	0.777050	0.388525	−	0.921438i	$-0.372985\pi$
$269$	12.7446	0.777050	0.388525	+	0.921438i	$0.372985\pi$
$270$	0	0
$271$	17.2554	1.04819	0.524097	−	0.851659i	$-0.324403\pi$
$271$	17.2554	1.04819	0.524097	+	0.851659i	$0.324403\pi$
$272$	0	0
$273$	−3.37228	−0.204100
$274$	0	0
$275$	0	0
$276$	0	0
$277$	24.9783	1.50080	0.750399	−	0.660985i	$-0.229861\pi$
$277$	24.9783	1.50080	0.750399	+	0.660985i	$0.229861\pi$
$278$	0	0
$279$	6.74456	0.403786
$280$	0	0
$281$	12.9783	0.774218	0.387109	−	0.922034i	$-0.373474\pi$
$281$	12.9783	0.774218	0.387109	+	0.922034i	$0.373474\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.62772	0.273166
$288$	0	0
$289$	−15.1168	−0.889226
$290$	0	0
$291$	−2.62772	−0.154040
$292$	0	0
$293$	15.2554	0.891232	0.445616	−	0.895224i	$-0.352985\pi$
$293$	15.2554	0.891232	0.445616	+	0.895224i	$0.352985\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.37228	−0.195680
$298$	0	0
$299$	0.627719	0.0363019
$300$	0	0
$301$	−13.4891	−0.777500
$302$	0	0
$303$	4.74456	0.272568
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−15.3723	−0.877342	−0.438671	−	0.898648i	$-0.644551\pi$
$307$	−15.3723	−0.877342	−0.438671	+	0.898648i	$0.644551\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.2337	1.02411	0.512053	−	0.858954i	$-0.328885\pi$
$317$	18.2337	1.02411	0.512053	+	0.858954i	$0.328885\pi$
$318$	0	0
$319$	6.74456	0.377623
$320$	0	0
$321$	8.62772	0.481552
$322$	0	0
$323$	−9.25544	−0.514986
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.74456	0.483575
$328$	0	0
$329$	−4.23369	−0.233411
$330$	0	0
$331$	−21.4891	−1.18115	−0.590575	−	0.806983i	$-0.701099\pi$
$331$	−21.4891	−1.18115	−0.590575	+	0.806983i	$0.701099\pi$
$332$	0	0
$333$	−5.37228	−0.294399
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.2337	0.993252	0.496626	−	0.867965i	$-0.334572\pi$
$337$	18.2337	0.993252	0.496626	+	0.867965i	$0.334572\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	−22.7446	−1.23169
$342$	0	0
$343$	8.86141	0.478471
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.8614	−1.11990	−0.559949	−	0.828527i	$-0.689179\pi$
$347$	−20.8614	−1.11990	−0.559949	+	0.828527i	$0.689179\pi$
$348$	0	0
$349$	−2.23369	−0.119567	−0.0597833	−	0.998211i	$-0.519041\pi$
$349$	−2.23369	−0.119567	−0.0597833	+	0.998211i	$0.519041\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.62772	−0.244925
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	26.4891	1.39416
$362$	0	0
$363$	0.372281	0.0195397
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9.48913	0.495328	0.247664	−	0.968846i	$-0.420337\pi$
$367$	9.48913	0.495328	0.247664	+	0.968846i	$0.420337\pi$
$368$	0	0
$369$	−1.37228	−0.0714381
$370$	0	0
$371$	−31.6060	−1.64090
$372$	0	0
$373$	15.2554	0.789897	0.394948	−	0.918703i	$-0.370763\pi$
$373$	15.2554	0.789897	0.394948	+	0.918703i	$0.370763\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	14.7446	0.757377	0.378689	−	0.925524i	$-0.376375\pi$
$379$	14.7446	0.757377	0.378689	+	0.925524i	$0.376375\pi$
$380$	0	0
$381$	16.2337	0.831677
$382$	0	0
$383$	−9.25544	−0.472931	−0.236465	−	0.971640i	$-0.575989\pi$
$383$	−9.25544	−0.472931	−0.236465	+	0.971640i	$0.575989\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	26.2337	1.33010	0.665050	−	0.746798i	$-0.268410\pi$
$389$	26.2337	1.33010	0.665050	+	0.746798i	$0.268410\pi$
$390$	0	0
$391$	0.861407	0.0435632
$392$	0	0
$393$	−18.7446	−0.945538
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−24.3505	−1.22212	−0.611059	−	0.791585i	$-0.709256\pi$
$397$	−24.3505	−1.22212	−0.611059	+	0.791585i	$0.709256\pi$
$398$	0	0
$399$	22.7446	1.13865
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	6.74456	0.335971
$404$	0	0
$405$	0	0
$406$	0	0
$407$	18.1168	0.898019
$408$	0	0
$409$	12.9783	0.641733	0.320867	−	0.947124i	$-0.396026\pi$
$409$	12.9783	0.641733	0.320867	+	0.947124i	$0.396026\pi$
$410$	0	0
$411$	−11.4891	−0.566717
$412$	0	0
$413$	26.9783	1.32751
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.37228	0.361022
$418$	0	0
$419$	−32.2337	−1.57472	−0.787359	−	0.616494i	$-0.788552\pi$
$419$	−32.2337	−1.57472	−0.787359	+	0.616494i	$0.788552\pi$
$420$	0	0
$421$	27.7228	1.35113	0.675564	−	0.737302i	$-0.263900\pi$
$421$	27.7228	1.35113	0.675564	+	0.737302i	$0.263900\pi$
$422$	0	0
$423$	1.25544	0.0610415
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−27.3723	−1.32464
$428$	0	0
$429$	−3.37228	−0.162815
$430$	0	0
$431$	18.9783	0.914150	0.457075	−	0.889428i	$-0.348897\pi$
$431$	18.9783	0.914150	0.457075	+	0.889428i	$0.348897\pi$
$432$	0	0
$433$	−6.23369	−0.299572	−0.149786	−	0.988718i	$-0.547858\pi$
$433$	−6.23369	−0.299572	−0.149786	+	0.988718i	$0.547858\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.23369	−0.202525
$438$	0	0
$439$	0.394031	0.0188061	0.00940303	−	0.999956i	$-0.497007\pi$
$439$	0.394031	0.0188061	0.00940303	+	0.999956i	$0.497007\pi$
$440$	0	0
$441$	4.37228	0.208204
$442$	0	0
$443$	−18.3505	−0.871860	−0.435930	−	0.899981i	$-0.643580\pi$
$443$	−18.3505	−0.871860	−0.435930	+	0.899981i	$0.643580\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	13.3723	0.632487
$448$	0	0
$449$	−1.37228	−0.0647620	−0.0323810	−	0.999476i	$-0.510309\pi$
$449$	−1.37228	−0.0647620	−0.0323810	+	0.999476i	$0.510309\pi$
$450$	0	0
$451$	4.62772	0.217911
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−40.1168	−1.87659	−0.938293	−	0.345840i	$-0.887594\pi$
$457$	−40.1168	−1.87659	−0.938293	+	0.345840i	$0.887594\pi$
$458$	0	0
$459$	1.37228	0.0640526
$460$	0	0
$461$	−20.3505	−0.947819	−0.473909	−	0.880574i	$-0.657158\pi$
$461$	−20.3505	−0.947819	−0.473909	+	0.880574i	$0.657158\pi$
$462$	0	0
$463$	31.6060	1.46885	0.734427	−	0.678688i	$-0.237451\pi$
$463$	31.6060	1.46885	0.734427	+	0.678688i	$0.237451\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.3505	−0.478965	−0.239483	−	0.970901i	$-0.576978\pi$
$467$	−10.3505	−0.478965	−0.239483	+	0.970901i	$0.576978\pi$
$468$	0	0
$469$	−13.4891	−0.622870
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	−13.4891	−0.620231
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.37228	0.429127
$478$	0	0
$479$	13.8832	0.634338	0.317169	−	0.948369i	$-0.397268\pi$
$479$	13.8832	0.634338	0.317169	+	0.948369i	$0.397268\pi$
$480$	0	0
$481$	−5.37228	−0.244955
$482$	0	0
$483$	−2.11684	−0.0963197
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−2.11684	−0.0959234	−0.0479617	−	0.998849i	$-0.515273\pi$
$487$	−2.11684	−0.0959234	−0.0479617	+	0.998849i	$0.515273\pi$
$488$	0	0
$489$	0.627719	0.0283864
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−2.74456	−0.123609
$494$	0	0
$495$	0	0
$496$	0	0
$497$	38.3505	1.72026
$498$	0	0
$499$	−14.7446	−0.660057	−0.330029	−	0.943971i	$-0.607058\pi$
$499$	−14.7446	−0.660057	−0.330029	+	0.943971i	$0.607058\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	22.9783	1.02455	0.512275	−	0.858822i	$-0.328803\pi$
$503$	22.9783	1.02455	0.512275	+	0.858822i	$0.328803\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	34.8614	1.54520	0.772602	−	0.634890i	$-0.218955\pi$
$509$	34.8614	1.54520	0.772602	+	0.634890i	$0.218955\pi$
$510$	0	0
$511$	−52.2337	−2.31068
$512$	0	0
$513$	−6.74456	−0.297780
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.23369	−0.186197
$518$	0	0
$519$	−7.48913	−0.328736
$520$	0	0
$521$	23.4891	1.02908	0.514539	−	0.857467i	$-0.327963\pi$
$521$	23.4891	1.02908	0.514539	+	0.857467i	$0.327963\pi$
$522$	0	0
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.25544	0.403173
$528$	0	0
$529$	−22.6060	−0.982868
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	−1.37228	−0.0594401
$534$	0	0
$535$	0	0
$536$	0	0
$537$	24.2337	1.04576
$538$	0	0
$539$	−14.7446	−0.635093
$540$	0	0
$541$	23.4891	1.00988	0.504938	−	0.863156i	$-0.331515\pi$
$541$	23.4891	1.00988	0.504938	+	0.863156i	$0.331515\pi$
$542$	0	0
$543$	−5.37228	−0.230547
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−6.97825	−0.298368	−0.149184	−	0.988809i	$-0.547665\pi$
$547$	−6.97825	−0.298368	−0.149184	+	0.988809i	$0.547665\pi$
$548$	0	0
$549$	8.11684	0.346418
$550$	0	0
$551$	13.4891	0.574656
$552$	0	0
$553$	56.8614	2.41799
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12.9783	−0.549906	−0.274953	−	0.961458i	$-0.588662\pi$
$557$	−12.9783	−0.549906	−0.274953	+	0.961458i	$0.588662\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	−4.62772	−0.195382
$562$	0	0
$563$	−1.88316	−0.0793656	−0.0396828	−	0.999212i	$-0.512635\pi$
$563$	−1.88316	−0.0793656	−0.0396828	+	0.999212i	$0.512635\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.37228	−0.141623
$568$	0	0
$569$	−23.7228	−0.994512	−0.497256	−	0.867604i	$-0.665659\pi$
$569$	−23.7228	−0.994512	−0.497256	+	0.867604i	$0.665659\pi$
$570$	0	0
$571$	−34.3505	−1.43753	−0.718763	−	0.695256i	$-0.755291\pi$
$571$	−34.3505	−1.43753	−0.718763	+	0.695256i	$0.755291\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−40.1168	−1.67009	−0.835043	−	0.550185i	$-0.814557\pi$
$577$	−40.1168	−1.67009	−0.835043	+	0.550185i	$0.814557\pi$
$578$	0	0
$579$	−9.37228	−0.389499
$580$	0	0
$581$	−40.4674	−1.67887
$582$	0	0
$583$	−31.6060	−1.30899
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.74456	0.113280	0.0566401	−	0.998395i	$-0.481961\pi$
$587$	2.74456	0.113280	0.0566401	+	0.998395i	$0.481961\pi$
$588$	0	0
$589$	−45.4891	−1.87435
$590$	0	0
$591$	10.2337	0.420958
$592$	0	0
$593$	11.2554	0.462205	0.231103	−	0.972929i	$-0.425767\pi$
$593$	11.2554	0.462205	0.231103	+	0.972929i	$0.425767\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	21.4891	0.879491
$598$	0	0
$599$	5.48913	0.224280	0.112140	−	0.993692i	$-0.464230\pi$
$599$	5.48913	0.224280	0.112140	+	0.993692i	$0.464230\pi$
$600$	0	0
$601$	−37.6060	−1.53398	−0.766990	−	0.641659i	$-0.778246\pi$
$601$	−37.6060	−1.53398	−0.766990	+	0.641659i	$0.778246\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	0	0
$609$	6.74456	0.273303
$610$	0	0
$611$	1.25544	0.0507896
$612$	0	0
$613$	14.8614	0.600247	0.300123	−	0.953900i	$-0.402972\pi$
$613$	14.8614	0.600247	0.300123	+	0.953900i	$0.402972\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−46.4674	−1.87071	−0.935353	−	0.353716i	$-0.884918\pi$
$617$	−46.4674	−1.87071	−0.935353	+	0.353716i	$0.884918\pi$
$618$	0	0
$619$	41.7228	1.67698	0.838491	−	0.544916i	$-0.183438\pi$
$619$	41.7228	1.67698	0.838491	+	0.544916i	$0.183438\pi$
$620$	0	0
$621$	0.627719	0.0251895
$622$	0	0
$623$	−45.0951	−1.80670
$624$	0	0
$625$	0	0
$626$	0	0
$627$	22.7446	0.908330
$628$	0	0
$629$	−7.37228	−0.293952
$630$	0	0
$631$	−18.5109	−0.736906	−0.368453	−	0.929646i	$-0.620113\pi$
$631$	−18.5109	−0.736906	−0.368453	+	0.929646i	$0.620113\pi$
$632$	0	0
$633$	22.9783	0.913303
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.37228	0.173236
$638$	0	0
$639$	−11.3723	−0.449880
$640$	0	0
$641$	22.2337	0.878178	0.439089	−	0.898444i	$-0.355301\pi$
$641$	22.2337	0.878178	0.439089	+	0.898444i	$0.355301\pi$
$642$	0	0
$643$	6.11684	0.241225	0.120612	−	0.992700i	$-0.461514\pi$
$643$	6.11684	0.241225	0.120612	+	0.992700i	$0.461514\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.86141	0.191122	0.0955608	−	0.995424i	$-0.469536\pi$
$647$	4.86141	0.191122	0.0955608	+	0.995424i	$0.469536\pi$
$648$	0	0
$649$	26.9783	1.05899
$650$	0	0
$651$	−22.7446	−0.891430
$652$	0	0
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	15.4891	0.604288
$658$	0	0
$659$	−14.5109	−0.565263	−0.282632	−	0.959229i	$-0.591207\pi$
$659$	−14.5109	−0.565263	−0.282632	+	0.959229i	$0.591207\pi$
$660$	0	0
$661$	−34.2337	−1.33154	−0.665768	−	0.746159i	$-0.731896\pi$
$661$	−34.2337	−1.33154	−0.665768	+	0.746159i	$0.731896\pi$
$662$	0	0
$663$	1.37228	0.0532950
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.25544	−0.0486107
$668$	0	0
$669$	−18.9783	−0.733742
$670$	0	0
$671$	−27.3723	−1.05670
$672$	0	0
$673$	8.51087	0.328070	0.164035	−	0.986455i	$-0.447549\pi$
$673$	8.51087	0.328070	0.164035	+	0.986455i	$0.447549\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.3723	0.667671	0.333836	−	0.942631i	$-0.391657\pi$
$677$	17.3723	0.667671	0.333836	+	0.942631i	$0.391657\pi$
$678$	0	0
$679$	8.86141	0.340070
$680$	0	0
$681$	18.7446	0.718293
$682$	0	0
$683$	−1.02175	−0.0390962	−0.0195481	−	0.999809i	$-0.506223\pi$
$683$	−1.02175	−0.0390962	−0.0195481	+	0.999809i	$0.506223\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.00000	0.0763048
$688$	0	0
$689$	9.37228	0.357055
$690$	0	0
$691$	−33.7228	−1.28288	−0.641438	−	0.767175i	$-0.721662\pi$
$691$	−33.7228	−1.28288	−0.641438	+	0.767175i	$0.721662\pi$
$692$	0	0
$693$	11.3723	0.431997
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.88316	−0.0713296
$698$	0	0
$699$	18.6277	0.704565
$700$	0	0
$701$	−16.7446	−0.632433	−0.316217	−	0.948687i	$-0.602413\pi$
$701$	−16.7446	−0.632433	−0.316217	+	0.948687i	$0.602413\pi$
$702$	0	0
$703$	36.2337	1.36658
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−16.0000	−0.601742
$708$	0	0
$709$	−39.2554	−1.47427	−0.737134	−	0.675746i	$-0.763822\pi$
$709$	−39.2554	−1.47427	−0.737134	+	0.675746i	$0.763822\pi$
$710$	0	0
$711$	−16.8614	−0.632352
$712$	0	0
$713$	4.23369	0.158553
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−22.3505	−0.834696
$718$	0	0
$719$	50.9783	1.90117	0.950584	−	0.310468i	$-0.100486\pi$
$719$	50.9783	1.90117	0.950584	+	0.310468i	$0.100486\pi$
$720$	0	0
$721$	−13.4891	−0.502361
$722$	0	0
$723$	−24.9783	−0.928951
$724$	0	0
$725$	0	0
$726$	0	0
$727$	10.7446	0.398494	0.199247	−	0.979949i	$-0.436150\pi$
$727$	10.7446	0.398494	0.199247	+	0.979949i	$0.436150\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.48913	0.203023
$732$	0	0
$733$	43.0951	1.59175	0.795877	−	0.605459i	$-0.207010\pi$
$733$	43.0951	1.59175	0.795877	+	0.605459i	$0.207010\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.4891	−0.496878
$738$	0	0
$739$	28.2337	1.03859	0.519296	−	0.854594i	$-0.326194\pi$
$739$	28.2337	1.03859	0.519296	+	0.854594i	$0.326194\pi$
$740$	0	0
$741$	−6.74456	−0.247768
$742$	0	0
$743$	12.2337	0.448810	0.224405	−	0.974496i	$-0.427956\pi$
$743$	12.2337	0.448810	0.224405	+	0.974496i	$0.427956\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	−29.0951	−1.06311
$750$	0	0
$751$	8.86141	0.323357	0.161679	−	0.986843i	$-0.448309\pi$
$751$	8.86141	0.323357	0.161679	+	0.986843i	$0.448309\pi$
$752$	0	0
$753$	17.4891	0.637339
$754$	0	0
$755$	0	0
$756$	0	0
$757$	39.2554	1.42676	0.713382	−	0.700776i	$-0.247163\pi$
$757$	39.2554	1.42676	0.713382	+	0.700776i	$0.247163\pi$
$758$	0	0
$759$	−2.11684	−0.0768366
$760$	0	0
$761$	−0.510875	−0.0185192	−0.00925960	−	0.999957i	$-0.502947\pi$
$761$	−0.510875	−0.0185192	−0.00925960	+	0.999957i	$0.502947\pi$
$762$	0	0
$763$	−29.4891	−1.06758
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	6.23369	0.224793	0.112396	−	0.993663i	$-0.464147\pi$
$769$	6.23369	0.224793	0.112396	+	0.993663i	$0.464147\pi$
$770$	0	0
$771$	−10.0000	−0.360141
$772$	0	0
$773$	−16.7446	−0.602260	−0.301130	−	0.953583i	$-0.597364\pi$
$773$	−16.7446	−0.602260	−0.301130	+	0.953583i	$0.597364\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	18.1168	0.649938
$778$	0	0
$779$	9.25544	0.331610
$780$	0	0
$781$	38.3505	1.37229
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	0	0
$786$	0	0
$787$	6.97825	0.248748	0.124374	−	0.992235i	$-0.460308\pi$
$787$	6.97825	0.248748	0.124374	+	0.992235i	$0.460308\pi$
$788$	0	0
$789$	17.4891	0.622629
$790$	0	0
$791$	−20.2337	−0.719427
$792$	0	0
$793$	8.11684	0.288238
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.6277	0.376453	0.188227	−	0.982126i	$-0.439726\pi$
$797$	10.6277	0.376453	0.188227	+	0.982126i	$0.439726\pi$
$798$	0	0
$799$	1.72281	0.0609488
$800$	0	0
$801$	13.3723	0.472486
$802$	0	0
$803$	−52.2337	−1.84329
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.7446	0.448630
$808$	0	0
$809$	34.4674	1.21181	0.605904	−	0.795538i	$-0.292811\pi$
$809$	34.4674	1.21181	0.605904	+	0.795538i	$0.292811\pi$
$810$	0	0
$811$	29.4891	1.03550	0.517752	−	0.855531i	$-0.326769\pi$
$811$	29.4891	1.03550	0.517752	+	0.855531i	$0.326769\pi$
$812$	0	0
$813$	17.2554	0.605175
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−26.9783	−0.943850
$818$	0	0
$819$	−3.37228	−0.117837
$820$	0	0
$821$	39.0951	1.36443	0.682214	−	0.731152i	$-0.261017\pi$
$821$	39.0951	1.36443	0.682214	+	0.731152i	$0.261017\pi$
$822$	0	0
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.2337	1.12088	0.560438	−	0.828197i	$-0.310633\pi$
$827$	32.2337	1.12088	0.560438	+	0.828197i	$0.310633\pi$
$828$	0	0
$829$	−20.9783	−0.728605	−0.364302	−	0.931281i	$-0.618693\pi$
$829$	−20.9783	−0.728605	−0.364302	+	0.931281i	$0.618693\pi$
$830$	0	0
$831$	24.9783	0.866486
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.74456	0.233126
$838$	0	0
$839$	−11.3723	−0.392615	−0.196307	−	0.980542i	$-0.562895\pi$
$839$	−11.3723	−0.392615	−0.196307	+	0.980542i	$0.562895\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	12.9783	0.446995
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.25544	−0.0431373
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	−3.37228	−0.115600
$852$	0	0
$853$	3.88316	0.132957	0.0664784	−	0.997788i	$-0.478824\pi$
$853$	3.88316	0.132957	0.0664784	+	0.997788i	$0.478824\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.6060	0.738046	0.369023	−	0.929420i	$-0.379692\pi$
$857$	21.6060	0.738046	0.369023	+	0.929420i	$0.379692\pi$
$858$	0	0
$859$	−30.1168	−1.02757	−0.513787	−	0.857918i	$-0.671758\pi$
$859$	−30.1168	−1.02757	−0.513787	+	0.857918i	$0.671758\pi$
$860$	0	0
$861$	4.62772	0.157712
$862$	0	0
$863$	−46.7446	−1.59120	−0.795602	−	0.605820i	$-0.792845\pi$
$863$	−46.7446	−1.59120	−0.795602	+	0.605820i	$0.792845\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−15.1168	−0.513395
$868$	0	0
$869$	56.8614	1.92889
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	−2.62772	−0.0889348
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.9783	−0.438244	−0.219122	−	0.975697i	$-0.570319\pi$
$877$	−12.9783	−0.438244	−0.219122	+	0.975697i	$0.570319\pi$
$878$	0	0
$879$	15.2554	0.514553
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	6.97825	0.234837	0.117418	−	0.993083i	$-0.462538\pi$
$883$	6.97825	0.234837	0.117418	+	0.993083i	$0.462538\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	50.3505	1.69061	0.845303	−	0.534288i	$-0.179420\pi$
$887$	50.3505	1.69061	0.845303	+	0.534288i	$0.179420\pi$
$888$	0	0
$889$	−54.7446	−1.83607
$890$	0	0
$891$	−3.37228	−0.112976
$892$	0	0
$893$	−8.46738	−0.283350
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0.627719	0.0209589
$898$	0	0
$899$	−13.4891	−0.449888
$900$	0	0
$901$	12.8614	0.428476
$902$	0	0
$903$	−13.4891	−0.448890
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−37.2554	−1.23705	−0.618523	−	0.785767i	$-0.712269\pi$
$907$	−37.2554	−1.23705	−0.618523	+	0.785767i	$0.712269\pi$
$908$	0	0
$909$	4.74456	0.157367
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	−40.4674	−1.33927
$914$	0	0
$915$	0	0
$916$	0	0
$917$	63.2119	2.08744
$918$	0	0
$919$	−48.8614	−1.61179	−0.805895	−	0.592059i	$-0.798315\pi$
$919$	−48.8614	−1.61179	−0.805895	+	0.592059i	$0.798315\pi$
$920$	0	0
$921$	−15.3723	−0.506534
$922$	0	0
$923$	−11.3723	−0.374323
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.00000	0.131377
$928$	0	0
$929$	10.3940	0.341017	0.170509	−	0.985356i	$-0.445459\pi$
$929$	10.3940	0.341017	0.170509	+	0.985356i	$0.445459\pi$
$930$	0	0
$931$	−29.4891	−0.966467
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.48913	0.113985	0.0569924	−	0.998375i	$-0.481849\pi$
$937$	3.48913	0.113985	0.0569924	+	0.998375i	$0.481849\pi$
$938$	0	0
$939$	6.00000	0.195803
$940$	0	0
$941$	37.3723	1.21830	0.609151	−	0.793054i	$-0.291510\pi$
$941$	37.3723	1.21830	0.609151	+	0.793054i	$0.291510\pi$
$942$	0	0
$943$	−0.861407	−0.0280513
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.9783	1.52659	0.763294	−	0.646051i	$-0.223581\pi$
$947$	46.9783	1.52659	0.763294	+	0.646051i	$0.223581\pi$
$948$	0	0
$949$	15.4891	0.502798
$950$	0	0
$951$	18.2337	0.591268
$952$	0	0
$953$	60.3505	1.95495	0.977473	−	0.211062i	$-0.0676921\pi$
$953$	60.3505	1.95495	0.977473	+	0.211062i	$0.0676921\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.74456	0.218021
$958$	0	0
$959$	38.7446	1.25113
$960$	0	0
$961$	14.4891	0.467391
$962$	0	0
$963$	8.62772	0.278024
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−34.9783	−1.12482	−0.562412	−	0.826857i	$-0.690127\pi$
$967$	−34.9783	−1.12482	−0.562412	+	0.826857i	$0.690127\pi$
$968$	0	0
$969$	−9.25544	−0.297327
$970$	0	0
$971$	6.51087	0.208944	0.104472	−	0.994528i	$-0.466685\pi$
$971$	6.51087	0.208944	0.104472	+	0.994528i	$0.466685\pi$
$972$	0	0
$973$	−24.8614	−0.797020
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−60.7446	−1.94339	−0.971695	−	0.236237i	$-0.924086\pi$
$977$	−60.7446	−1.94339	−0.971695	+	0.236237i	$0.924086\pi$
$978$	0	0
$979$	−45.0951	−1.44125
$980$	0	0
$981$	8.74456	0.279192
$982$	0	0
$983$	−5.48913	−0.175076	−0.0875380	−	0.996161i	$-0.527900\pi$
$983$	−5.48913	−0.175076	−0.0875380	+	0.996161i	$0.527900\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−4.23369	−0.134760
$988$	0	0
$989$	2.51087	0.0798412
$990$	0	0
$991$	−15.6060	−0.495740	−0.247870	−	0.968793i	$-0.579731\pi$
$991$	−15.6060	−0.495740	−0.247870	+	0.968793i	$0.579731\pi$
$992$	0	0
$993$	−21.4891	−0.681937
$994$	0	0
$995$	0	0
$996$	0	0
$997$	23.7228	0.751309	0.375655	−	0.926760i	$-0.377418\pi$
$997$	23.7228	0.751309	0.375655	+	0.926760i	$0.377418\pi$
$998$	0	0
$999$	−5.37228	−0.169971

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bb.1.1		2
5.4	even	2		1560.2.a.n.1.2	✓	2
15.14	odd	2		4680.2.a.bf.1.2		2
20.19	odd	2		3120.2.a.bd.1.1		2
60.59	even	2		9360.2.a.co.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.a.n.1.2	✓	2	5.4	even	2
3120.2.a.bd.1.1		2	20.19	odd	2
4680.2.a.bf.1.2		2	15.14	odd	2
7800.2.a.bb.1.1		2	1.1	even	1	trivial
9360.2.a.co.1.1		2	60.59	even	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.37228 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.37228 q^{7} +1.00000 q^{9} -3.37228 q^{11} +1.00000 q^{13} +1.37228 q^{17} -6.74456 q^{19} -3.37228 q^{21} +0.627719 q^{23} +1.00000 q^{27} -2.00000 q^{29} +6.74456 q^{31} -3.37228 q^{33} -5.37228 q^{37} +1.00000 q^{39} -1.37228 q^{41} +4.00000 q^{43} +1.25544 q^{47} +4.37228 q^{49} +1.37228 q^{51} +9.37228 q^{53} -6.74456 q^{57} -8.00000 q^{59} +8.11684 q^{61} -3.37228 q^{63} +4.00000 q^{67} +0.627719 q^{69} -11.3723 q^{71} +15.4891 q^{73} +11.3723 q^{77} -16.8614 q^{79} +1.00000 q^{81} +12.0000 q^{83} -2.00000 q^{87} +13.3723 q^{89} -3.37228 q^{91} +6.74456 q^{93} -2.62772 q^{97} -3.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - q^7 + 2 * q^9	\( 2 q + 2 q^{3} - q^{7} + 2 q^{9} - q^{11} + 2 q^{13} - 3 q^{17} - 2 q^{19} - q^{21} + 7 q^{23} + 2 q^{27} - 4 q^{29} + 2 q^{31} - q^{33} - 5 q^{37} + 2 q^{39} + 3 q^{41} + 8 q^{43} + 14 q^{47} + 3 q^{49} - 3 q^{51} + 13 q^{53} - 2 q^{57} - 16 q^{59} - q^{61} - q^{63} + 8 q^{67} + 7 q^{69} - 17 q^{71} + 8 q^{73} + 17 q^{77} - 5 q^{79} + 2 q^{81} + 24 q^{83} - 4 q^{87} + 21 q^{89} - q^{91} + 2 q^{93} - 11 q^{97} - q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - q^7 + 2 * q^9 - q^11 + 2 * q^13 - 3 * q^17 - 2 * q^19 - q^21 + 7 * q^23 + 2 * q^27 - 4 * q^29 + 2 * q^31 - q^33 - 5 * q^37 + 2 * q^39 + 3 * q^41 + 8 * q^43 + 14 * q^47 + 3 * q^49 - 3 * q^51 + 13 * q^53 - 2 * q^57 - 16 * q^59 - q^61 - q^63 + 8 * q^67 + 7 * q^69 - 17 * q^71 + 8 * q^73 + 17 * q^77 - 5 * q^79 + 2 * q^81 + 24 * q^83 - 4 * q^87 + 21 * q^89 - q^91 + 2 * q^93 - 11 * q^97 - q^99

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